Question

How do you factor \[9{{x}^{2}}-49\]?

Answer 1

Hint: For the given question we are asked to find the factor \[9{{x}^{2}}-49\]. For this question we will bring the given question into \[{{x}^{2}}-{{y}^{2}}\] form and use the basic algebra and its formula which is \[{{x}^{2}}-{{y}^{2}}=\left( x-y \right)\left( x+y \right)\] and simplify the given question and find the required solution to the given question.

Complete step by step solution:
Firstly, as mentioned in the above will try and bring the given question into \[{{x}^{2}}-{{y}^{2}}\] form.
So, we get the expression as follows.
\[\Rightarrow 9{{x}^{2}}-49\]
We know that \[49\] is a square of integer \[7\] and we can write 9 as a square of integer 3. So, the expression will be reduced as follows.
\[\Rightarrow {{\left( 3x \right)}^{2}}-{{7}^{2}}\]
So, after reducing the expression into \[{{x}^{2}}-{{y}^{2}}\] form we will use the basic algebraic formula which is \[{{x}^{2}}-{{y}^{2}}=\left( x-y \right)\left( x+y \right)\] and simplify further to get the required solution.
So, after using the basic algebraic formula we get the expression reduced as follows.
Here we compared that \[y=7\] and x as \[3x\]. So, after substituting the values we got in the basic algebraic formula we get the expression reduced as follows.
\[\Rightarrow {{\left( 3x \right)}^{2}}-{{7}^{2}}\]
\[\Rightarrow \left( 3x+7 \right)\left( 3x-7 \right)\]
Therefore, the solution for the given question will be as follows.
\[\Rightarrow \left( 3x+7 \right)\left( 3x-7 \right)\]

Note:
Students must be very careful while solving questions of this kind and students must be having good knowledge in the concept of algebra and must know its formulae very well. We can check the solution by the quadratic equations concept as follows. The root of a quadratic equation will be \[\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]. So, from this formula \[\dfrac{-0\pm \sqrt{^{{}}-4\left( 9 \right)\left( -49 \right)}}{2\times 9}=\dfrac{7}{3}\].
So, the factor will be \[\Rightarrow \left( 3x+7 \right)\left( 3x-7 \right)\].